Problem 91 - A Complete Resource Book in Mathematics for JEE Main

Question

The equation \(r=|\cos \theta|\) represents (A) two circles of radii \(\frac{1}{2}\) each (B) two circles centered at \(\left(\frac{1}{2}, 0\right)\) and \(\left(-\frac{1}{2}, 0\right)\) (C) two circles touching each other at the origin (D) pair of straight lines

Step-by-Step Solution

Answer

Option (C) is correct; the circles touch at the origin.

Step 1: Understand the Equation

The given polar equation is \(r = |\cos \theta|\). This equation describes points whose distance from the origin is the absolute value of the cosine of the angle \(\theta\).

Step 2: Analyze the Interval for Cosine

Recall that the cosine function \(\cos \theta\) varies between -1 and 1. Therefore, \(|\cos \theta|\) will vary between 0 and 1 because the absolute value function makes all negative values positive.

Step 3: Convert to Cartesian Coordinates

The polar equation can be represented in Cartesian coordinates: \(r = x^2 + y^2\) and \( \cos \theta = \frac{x}{\sqrt{x^2+y^2}} \). Rearranging yields the equation \(x^2 + y^2 = |x|\), considering \(r = |\cos \theta| = |x| / \sqrt{x^2+y^2}\).

Step 4: Simplify the Cartesian Equation

For \(x \geq 0\), the equation \(x^2 + y^2 = x\) represents a circle centered at \((\frac{1}{2}, 0)\) with radius \(\frac{1}{2}\). Similarly, for \(x < 0\), \(x^2 + y^2 = -x\) describes another circle centered at \((-\frac{1}{2}, 0)\) also with radius \(\frac{1}{2}\).

Step 5: Identify the Touching Point

Both circles are tangent to the y-axis at the origin \((0,0)\). They touch each other at this point. Therefore, the statement (C) is verified.

Key Concepts

Polar Equations

Polar equations describe the relationship between a point's distance from a central origin and the angle it forms with a reference direction, typically the positive x-axis. In polar coordinates, every point on the plane is specified by a radius and an angle.
For example, the equation \( r = |\cos \theta| \) describes a set of points where the distance \( r \) from the origin is the absolute value of the cosine of the angle \( \theta \). The absolute value ensures that \( r \) is always non-negative, which is crucial for defining radial distances.

Understanding and analyzing these equations often involves recognizing the behavior of trigonometric functions over their intervals.

• Cosine values range between -1 and 1.
• The absolute value of these values ranges between 0 and 1.

Polar equations might look complex, but they beautifully encode circular and angular relationships, making them valuable tools in geometry and many applied sciences.

Cartesian Coordinates

Converting polar equations into Cartesian coordinates can provide more familiar forms of geometric shapes. Cartesian coordinates use \(x\) and \(y\) to describe a point's horizontal and vertical distances from the origin. This system makes understanding geometric figures intuitive and straightforward.

When translating from polar to Cartesian coordinates, you often rely on relationships such as \(x = r \cos \theta\) and \(y = r \sin \theta\). These conversions open up polar equations to algebraic manipulations and recognitions of common geometric shapes. For the equation \( r = |\cos \theta| \), when rewritten in Cartesian form, you can use the identities:

• \( r = \sqrt{x^2 + y^2} \)
• \( \cos \theta = \frac{x}{\sqrt{x^2 + y^2}} \)

By equating \( r = |x|/ \sqrt{x^2 + y^2} \), the result simplifies to the equations \( x^2 + y^2 = x \) for \( x \geq 0 \) and \( x^2 + y^2 = -x \) for \( x < 0 \). These describe specific circle arrangements in the Cartesian plane.

Circle Geometry

Understanding the geometry of circles is essential when analyzing line and plane equations. Circles are defined by their center point and radius.

In Cartesian coordinates, a circle's equation typically takes the form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. By examining such equations, you can determine important information about the circle's placement and size.

For the given equations resulting from the polar conversion, \(x^2 + y^2 = x\) and \(x^2 + y^2 = -x\) form two circles:

• Center at \((1/2, 0)\) with radius \(1/2\)
• Center at \((-1/2, 0)\) also with radius \(1/2\)

These circles are tangent to each other at the origin \((0, 0)\), where they touch but do not overlap. Such geometry provides insights into spatial relationships and is crucial for problems in fields like physics, engineering, and computer graphics.